Quadrilaterals

General Quadrilaterals

Any quadrilateral shape can be used to tile a plane area without leaving gaps anywhere. The tiles occur in two orientations related by a half-turn.

The lines joining the opposite corners of a quadrilateral are its diagonals. They intersect internally if and only if the quadrilateral is convex.

The lines joining the midpoints of opposite sides and the line joining the midpoints M, N of the diagonals (if it exists) are concurrent. This point is the centroid of the quadrilateral. The line MN is the locus of points P such that area (PAB + PCD) = area (PAC + PBD).
AB² + BC² + CD² + DA² = AC² + BD² + 4MN² and AC² + BD² = 2(EG² + HF²).

The centroids of the four triangles formed by the vertices of a quadrilateral taken three at a time form a quadrilateral antithetic to the given quadrilateral (i.e. of the same shape and with the edges parallel but rotated 180°). The point of intersection of the four lines joining the corresponding vertices of the two quadrilaterals (their centre of similitude) is the centroid.


If squares are drawn outwards on the sides of a quadrilateral ABCD the centres of these squares form a second quadrilateral WXYZ. The diagonals of this quadrilateral are equal in length and at right angles to each other. This remains true if one of the sides is of zero length, i.e. the quadrilateral reduces to a triangle. [Aubel's theorem, Wells p.11] The full squares are not shown in the figure only the mediators.

If the squares are constructed inwards the same property is true.These two shorter segments and the longer pair have only two mid-points between them. The midpoint of the line joining them is the centroid of the four vertices of the original quadrilateral.

The mid-points of the four diagonals form a square. [S.Collings, IMA Bulletin, Nov-Dec 1984, item EFG-50]


Parallelograms

The straight line segments which join the corresponding ends of two equal and parallel straight line segments are themselves equal and parallel. They form a parallelogram.


Two sets of equidistant parallel lines crossing each other dissect the plane into a tesselation of congruent parallelograms.

The diagonals of a quadrilateral bisect each other if and only if it is a parallelogram.

A third set of equidistant parallels can be superimposed on the two given to dissect the plane into a tessellation of congruent triangles. The third set of parallels are diagonals of the parallelograms formed by the other two, i.e. they join opposite corners. Any one of the three sets can be seen as diagonals of the pattern formed by the other two.

The mid-points of the sides of a quadrilateral are the vertices of a parallelogram whose sides are parallel to the diagonals of the quadrilateral and half their length. Its area is half that of the quadrilateral in the convex case. The four pieces cut off the corners by the inscribed parallelogram in the convex case will fit together to form another parallelogram congruent to it.

The joins of the midpoints of opposite sides of a convex quadrilateral divide it into four parts that will fit together to form a parallelogram. If the four pieces are hinged at three of the midpoints then the transformation can be accomplished by opening out the chain of four pieces and closing it up in the opposite direction.

Parallelograms on the same base and between the same parallels are equal in area. When the parallelograms are not too steeply slanted with respect to each other the diagram for this proposition provides a means of transforming one parallelogram into another, or into a rectangle of the same height and area, by a two-part dissection, by cutting a piece off at one end and moving it to the other. The new parallelogram can then be similarly transformed, between the other parallels, so that a third parallelogram is formed that has different sides and angles to the original and is formed from it by a three-piece dissection. The new and old parallelograms can be arranged to form a repeating pattern, each of which tiles the entire plane.

Rectangles

A fourth set of equidistant parallels can be superimposed on the set of three, to divide then plane into a tessellation of two types of triangles. These two types become one type in the special case when the initial pair of parallels are mutually perpendicular. In this case the parallelograms are rectangles.

Parallelogons

Any even polygon with opposite sides parallel is a parallelogon. A 4-sided one is a parallelogram. If opposite sides are equal as well we have an equi-parallelogon. Parallelograms are 4-sided equiparallelogons.

Any convex equiparallelogon can be dissected into parallelograms.